Paul, (01)
The lattice of theories is nothing more nor less than a
mathematical technique that has some potential applications. (02)
> I am not rejecting the notion of a lattice of theories
> ordered by some formal method, but it does not seem to me
> that this lattice, as commonly understood, is going to
> allow users (business or medical community) to define
> what they (as the domain specialists) want to define and
> then let the lattice manage the merge and comparison issues. (03)
The lattice, by itself, doesn't solve anything. It is like
any mathematical technique. Every electrical engineer has to
study calculus, but calculus won't solve the problems. The
engineers have to do a lot of work to formulate the problems
in such a way that their mathematical techniques can be
applied. (04)
That is also true of any formalism of any kind. The formal
techniques require some study, but in the end, they are much,
much simpler than the really challenging problems of deciding
how and whether to use them and what to do with the results,
if any, of using them. (05)
On a related point, I sent the following message to another
forum in reply to some questions about the lattice of theories. (06)
John Sowa
____________________________________________________________ (07)
> You could classify collection systems in many ways... (08)
Of course. (09)
> Empty set, as it is in ZF, causes misunderstandings and
> is conceptually incoherent. (010)
I agree that it causes confusion among students. But it is
a perfectly reasonable mathematical assumption. You are free
to say that you don't like it, and use something else. (011)
But please note that I explicitly said that the infinite
lattice contains all the elegant theories and all the truly
ugly theories. You may have a different opinion about what
theories are elegant or ugly, but that opinion is irrelevant
to the question of where those theories are situated in the
lattice. (012)
> I'd put mereology under the category of Boolean algebras,
> and all collection formalisms that have the memberOf
> operator under the category Set theories. (013)
Actually, sets form a Boolean algebra with the empty set
corresponding to falsehood. Those versions of mereology
that don't have a null or empty element don't have a
natural analog of F. (014)
> I don't quite understand what you mean by a closure of
> axioms. (015)
That is standard terminology in logic, and I defined it in
the theories.htm paper: (016)
closure(S) = {p  p is provable from S] (017)
The closure of any set S of axioms is nothing more nor less
than the set of all propositions that are provable from S. (018)
> That sort of a belief [reflexivity] is similar to the belief
> that a man washes himself.... (019)
Some relations are reflexive and some aren't. If you like the
axiom, accept it. If you don't, ignore it. Preferences may
determine what axioms you decide to use, but they have no effect
on the structure of the lattice. (020)
> D.M.Armstrong's Theory of Universals 1978 19.VI:
>
> "One hand washes another; both wash the rest of the body.
> Perhaps the trickiest sort of case is that where a person
> loves or hates himself. But even here genuine selfrelation
> seems avoidable. If a man loves himself, then it is not
> that selfloving state which he loves, but other aspect
> of himself. It is possible that he should love the self
> loving state, but this seems to demand a new, second
> order, loving state which is distinct from the original one." (021)
That's an interesting solution to the apparent paradox of
loving and hating being antonyms. You could also solve that
problem by saying that loving and hating are not true antonyms.
In any case, that's irrelevant to the structure of the lattice. (022)
> BOT describes a contradiction, a round square. (023)
Yes. It includes all possible contradictions. (024)
> ... but in that case [a plan for something that doesn't yet
> exist] the category would describe something that exists inside
> the brains of a human being. And if that person dies or forgets
> what the category describes, the category would be like a circle
> in sand in some deserted island. (025)
What's wrong with categories that describe what's on a deserted
island? I believe that a tree that falls makes exactly the same
vibration in the air whether or not anyone is around to hear it.
Whether a mathematical theory is known to have many applications
or none at all has no effect on its status in the lattice. (026)
>> The absurd theory at the bottom is necessary to complete the
>> lattice. It is like the number 0, which counts nothing. (027)
> BOT is very problematic: you cannot divide by 0. (028)
So what? The problem only exists if you try to do that.
The solution is not to do that. (029)
John (030)
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